Title: A density result for universal quadratic forms over totally real quadratic number fields
Speaker: 박다윤 (UNIST)
Abstract: In this talk, I will introduce the recent results that
(1) there exist only finitely many real quadratic fields which admit a positive definite septenary universal quadratic lattices of rank $\leq 7$.
(2) and density of infinitely many real quadratic number fields which admit a positive definite universal quadratic lattices of rank $r$ for each given $r\geq 8$ is zero.
More strongly, we see similar result for quadratic lattices that represent all elements of $m\mathcal{O}^{+}$ (for fixed positive integer $m\in\mathbb{N}$).
This is a joint work with Vítězslav Kala, Pavlo Yatsyna, and Błażej Żmija.
Title: Cyclotomic Iwasawa $\mu$-invariant of real quadratic fields
Speaker: 권재성 (UNIST)
Abstract: Greenberg's conjecture is one of the important conjectures in Iwasawa theory, which is about the vanishing of the invariant of the Iwasawa theoretic objects. For example, it is proved by Ferrero-Washington that the $\mu$-invariant of abelian number fields is vanishing. In this talk, we construct a $p$-adic $L$-function by using the homological method, which interpolates the Hecke $L$-values over real quadratic fields. Also, we give the homological approach to prove the vanishing of cyclotomic $\mu$-invariant of real quadratic fields, which is a different method with the Ferrero-Washington's one. This is a joint work with Jungyun Lee and Hae-Sang Sun.
Title: Norm factorization formula for differences of CM values of Hauptmoduls
Speaker: 권영욱 (UNIST)
Abstract: Singular moduli are the values of the $j$-function at CM points. In 1985, Gross and Zagier considered the norms of differences of singular moduli and discovered factorization formulas for those values. Their result was reproved by Schofer in 2006. Schofer's idea is to use Borcherds lifting and the CM value formula. In this talk, I briefly will introduce Borcherds product and the CM value formula and I will report the progress in the project generalizing the Gross-Zagier factorization formula to Hauptmoduls for Atkin-Lehner groups of genus zero.
Title: Complete bilinear forms
Speaker: 박준영 (UNIST)
Abstract: In this talk, I will discuss a nice compactification of the space of bilinear forms of rank at least r, whose points are called complete bilinear forms. If time permits, I will say something on the ongoing joint work with Wansu Kim about the degeneration of shktukas.
Title: Frobenius eigenvalues in a space of mod-p automorphic forms
Speaker: 박철 (UNIST)
Abstract: For a given mod-p local Galois representation, one can construct a space of mod-p automorphic forms. One believes that this space is a candidate of the automorphic representation corresponding to the given mod-p local Galois representation for mod-p Langlands program. It is believed that Frobenius eigenvalues of certain potentially crystalline lifts of the given mod-p local Galois representation capture the extension classes of the given mod-p local Galois representation. In this talk, we introduce the actions of U_p-operators and discuss how one can use those U_p-operators to capture the Frobenius eigenvalues in the space of mod-p automorphic forms.
Title: Root numbers of CM elliptic curves
Speaker: 이완 (UNIST)
Abstract: Hasse-Weil conjecture asserts that L-function of a given elliptic curve over a number field has an analytic continuation to whole complex numbers and satisfies functional equation. The root number is defined to be the sign appearing in the functional equation. Although this definition is conjectural, there is another definition of Langlands and Deligne which is not conjectural. In this talk, we discuss local and global root number formula of CM elliptic curves. Using this formula, we can find infinitely many elliptic curves such that the global root numbers become 1 for all quadratic extensions (in which Dokchitser and Dokchitser calls 'lawful' elliptic curves). This is joint work with Myungjun Yu.